4.11. Appendices

4.11.1. Appendix 4.1: Steady-State Power Normalization in SAS4A/SASSYS‑1

The power in axial fuel pin segment \(I\) of channel \(J\) is given by the expression

\[{\text{PSHAPE}}\left( I,J \right)\ \times \ {\text{QMULT}} \times \ {\text{POW}}\]

Here POW is the maximum power of any axial fuel pin segment. QMULT is a multiplier that is equal to one in steady state.

\({\text{PSHAPE}}\left( I,J \right)\) is the ratio of the power of segment \(\left( I,J \right)\) to the maximum power of any segment. It is obvious that PSHAPE takes values between zero and one.

SAS4A/SASSYS‑1 will renormalize the input values of PSHAPE and PRSHAP. This is done is subroutine PNORM. The user needs only supply un-normalized data for these arrays.

PNORM will also compute POW from POWTOT or POWTOT from POW according to the value of IPOWOP.

Input:

IPOWOP = 0	Calculate steady-state power in peak axial segment from total reactor power.
IPOWOP = 1	Calculate steady-state total reactor power from the peak axial fuel pin segment.
POWTOT	Total reactor power in steady state.
POW	Steady-state power in the peak axial pin segment.
FRPR	Fraction of total reactor power represented by sum of all SAS4A/SASSYS‑1 channels.
NPIN(ICH)	Number of fuel pins in a subassembly of channel ICH.
NSUBAS(ICH)	Number of subassemblies in channel ICH.
MZ(ICH)	Number of axial nodes in channel ICH.
PRSHAP(ICH)	The relative power per subassembly in channel ICH. PRSHAP will be normalized by PNORM.
PSHAPE(I,ICH)	The relative power per axial pin segment of axial pin segment I and channel ICH. I = 1, … MZ(ICH). PSHAPE will be normalized by the PNORM routine.

Method:

The values of PRSHAP get renormalized as RELCHA for all channels:

\[{\text{RELCHA}}\left( \text{ICH} \right) = {\text{PRSHAP}}\left( \text{ICH} \right)\ \times \frac{\sum_{\text{I}}{{\text{NSUBAS}}\left( I \right)}}{\sum_{\text{I}}{{\text{PRSHAP}}\left( I \right) \times {\text{NSUBAS}}\left( I \right)}}\]

If we multiply the above equation by \({\text{NSUBAS}}\left( \text{ICH} \right)\) and then sum over all channels \(\text{ICH}\), we can show that

\[\sum_{\text{ICH}}{{\text{RELCHA}}\left( \text{ICH} \right)\ \times \ {\text{NSUBAS}}\left( \text{ICH} \right)} = \sum_{\text{ICH}}{{\text{NSUBAS}}\left( \text{ICH} \right)}\]

Thus, RELCHA is properly normalized.

The values of PSHAPE get normalized as RELSHP for each channel:

\[{\text{RELSHP}}\left( IZ,ICH \right) = \frac{{\text{PSHAPE}}\left( IZ,ICH \right)}{\sum_{\text{I}}{{\text{PSHAPE}}\left( I,ICH \right)}}\]

If we sum over all axial nodes \(\text{IZ}\), we can show that

\[\sum_{\text{IZ}}{{\text{RELSHP}}\left( IZ,ICH \right) = 1}\]

Thus, RELSHP is properly normalized.

The power of axial pin segment \(\left( I,J \right)\) is

\[{\text{RELSHP}}\left( I,J \right)\ \times \frac{{\text{RELCHA}}\left( J \right)}{{\text{NPIN}}\left( J \right)}\ \times \ \frac{{\text{FRPR}}\ \times \ {\text{POWTOT}}}{\sum_{\text{ICH}}{{\text{NSUBAS}}\left( \text{ICH} \right)}}\]

From the definition of POW we get the equation

\[{\text{POW}} = {\left\lbrack {\text{RELSHP}}\left( I,J \right)\ \times \frac{{\text{RELCHA}}\left( J \right)}{{\text{NPIN}}\left( J \right)}\ \right\rbrack \times \ \frac{{\text{FRPR}}\ \times \ {\text{POWTOT}}}{\sum_{\text{ICH}}{{\text{NSUBAS}}\left( \text{ICH} \right)}}}\]

and the unknown POWTOT or POW can be found. Given \(I_{\mathrm{\max}}\) and \(J_{\mathrm{\max}}\) to be the indices of the peak power pin segment, we then redefine \({\text{PSHAPE}}\left( I,J \right)\) as

\[{\text{PSHAPE}}\left( I,J \right) = \frac{{\text{RELSHP}}\left( I,J \right)\ \times \ {\text{RELCHA}}\left( J \right)}{{\text{NPIN}}\left( J \right)} \times \frac{{\text{NPIN}}\left( J_{\mathrm{\max}} \right)}{{\text{RELSHP}}\left( I_{\mathrm{\max}},J_{\mathrm{\max}} \right)\ \times \ {\text{RELCHA}}\left( J_{\mathrm{\max}} \right)}\]

This is the desired form of PSHAPE that is used by the subroutine SHAPE.